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OPTICS LETTERS / Vol. 38, No. 6 / March 15, 2013
Metamaterials-based broadband generation of orbital angular momentum carrying vector beams Zhe Zhao,1 Jian Wang,1,* Shuhui Li,1 and Alan E. Willner2 1
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Hubei, Wuhan 430074, China 2 Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA *Corresponding author:
[email protected] Received December 26, 2012; accepted January 27, 2013; posted February 11, 2013 (Doc. ID 182338); published March 13, 2013 We propose a simple approach to broadband generation of orbital angular momentum (OAM) carrying vector beams based on compact metamaterials. It consists of two concentric rings in a gold film, where each ring is composed of subwavelength rectangular apertures with gradually varied orientation. The subwavelength rectangular aperture serves as a localized spatial polarizer. We show the generation of different OAM-carrying vector beams with OAM charge number and polarization order varying from −3 to 3 using a 11.2 × 11.2 μm device. The extinction ratio can exceed 20 dB, and the operation bandwidth (1500 nm) can cover from 1000 to 2500 nm (from near-infrared to mid-infrared). The device provides three degrees of freedom (polarization order l, polarization of input beam σ, and initial orientation angle α0 ) to flexibly generate different OAM-carrying vector beams. We can use a single device to generate two OAM-carrying vector beams with opposite charge sign of OAM by simply controlling the polarization of the input beam. We further study the performance dependence of the designed metamaterials on the offset of the initial orientation angle, length, and width of the rectangular apertures. The obtained results indicate favorable fabrication tolerance. © 2013 Optical Society of America OCIS codes: 050.4865, 160.3918.
It is well known that phase and polarization are two important properties of electromagnetic waves. Manipulating phase and polarization in the local spatial domain can create so-called “vortex beams,” which have recently attracted increasing interest. Typically, there are two types of vortex beams. One class of vortex beams with phase singularity is featured by a helical phasefront of expimφ, where m is the topological charge and φ is the azimuthal angle [1]. Such “twisted” beams (helical phasefront) carry orbital angular momentum (OAM) of mħ per photon. OAM-carrying vortex beams have seen wide applications in optical manipulation, imaging, and optical communications [2–5]. Another class of vortex beams with polarization singularity is characterized by spatially variant polarization of αφ lφ α0 , where l is the polarization order and α0 is the initial polarization orientation for φ 0 [6]. Such “vector” beams (spatially variant polarization) have many applications in optical trapping, spectroscopy, and super-resolution microscopy [7]. Moreover, OAM-carrying vector beams, which combine both “twisted” and “vector” features, can give accesses to the spatial phase and polarization and might provide more degrees of freedom for beam manipulation. Driven by their distinctive properties and miscellaneous applications, there have been many attempts to generate OAM-carrying beams or vector beams. For instance, OAM-carrying beams have been generated by cylindrical lens mode converters, fiber mode couplers, patterned polarizers, q-plates, spiral phase plates, spatial light modulators, and silicon-integrated angular gratings [2,3,8–12]. Meanwhile, vector beams have also been achieved using different methods, such as laser intracavity devices, interferometry, and space-variant subwavelength gratings [7,13]. Despite successful generation of OAM-carrying beams or vector beams, one may see several challenges: (i) the employed devices are bulky and cumbersome, (ii) the generation offers a narrow bandwidth, and 0146-9592/13/060932-03$15.00/0
(iii) less attention is paid to control of both spatial phase and polarization. In this Letter, we design compact metamaterials to enable broadband generation of OAM-carrying vector beams. OAM-carrying vector beams with OAM charge number and polarization order from −3 to 3 are generated using a 11.2 × 11.2 μm device. We characterize the performance in terms of extinction ratio (ER), purity, operation bandwidth, design degrees of freedom, and fabrication tolerance. The presented metamaterials show flexible generation of OAM-carrying vector beams with high ER above 20 dB, broad bandwidth (1500 nm) from 1000 to 2500 nm (i.e., from near-infrared to mid-infrared), and favorable fabrication tolerance. We choose right or left circularly polarized light with Jones vector of Ein 1 iσ T (σ −1 or 1) as the input beam to generate an OAM-carrying vector beam. Here, we introduce two orthogonal unit vectors, e1 φ cos αφ sin αφ T and e2 φ sin αφ − cos αφT , where αφ lφ α0 (l is the polarization order, and α0 is the initial polarization orientation for φ 0). Then Ein can be rewritten as Ein eiσα0 eilσφ e1 φ − iσe2 φ:
(1)
Ignoring the constant phase of eiσα0 and using a spatially variant polarizer to select one of the two orthogonal components e1 φ [or e2 φ], we can get an OAM-carrying vector beam expressed as coslφ α0 ; sinlφ α0 − sinlφ α0 ilσφ ilσφ −iσe e2 φ iσe : coslφ α0
Eout eilσφ e1 φ eilσφ
Eout
(2a) (2b)
Equations (2a) and (2b) indicate that spatially variant polarizers [along e1 φ or e2 φ] can convert circularly © 2013 Optical Society of America
March 15, 2013 / Vol. 38, No. 6 / OPTICS LETTERS
polarized light to an OAM-carrying vector beam that has both OAM with a charge number of m lσ and spatial polarization with an order of l. Figures 1(a) and 1(b) present the schematic structure and geometric parameters of the proposed metamaterials. We design two concentric rings in a gold film with a thickness of h 200 nm. Each ring is composed of 42 rectangular apertures with gradually varied orientation. The rectangular aperture array can enhance the transmission of linearly polarized light (perpendicular to the aperture direction) owing to the excitation of waveguide resonance in the rectangular aperture [14,15]. Therefore, each rectangular aperture can be regarded as a localized linear polarizer. By controlling the orientation angle of rectangular apertures, we can construct desired spatially variant polarizers to generate OAM-carrying vector beams according to Eqs. (2a) and (2b). Figure 1(c) depicts the evolution of phasefront and spatial polarization after passing though the metamaterials, which indicates generation of an OAM-carrying vector beam from a circularly polarized beam. Figure 2 shows spatial distributions of phase, power, and polarization of output beams under excitation of input left circularly polarized light (Ein 1 i T ). We first focus on the wavelength at 1550 nm. The employed metamaterials have an orientation angle of αφ lφ α0 , where l varies from 3 to −3 and α0 0. We use E1 and E2 to represent the electric field components along directions of e1 φ and e2 φ, respectively. The first row of Fig. 2 shows spatial phase distribution of E1 , indicating that output beams carry OAM with a charge number of l from 3 to −3. The second and third rows of Fig. 2 show spatial power distributions P 1 ∝ jE1 j2 ; P 2 ∝ jE2 j2 along directions of e1 φ and e2 φ, respectively. The ER, defined by 10×log10 P 1∕P 2 , exceeds 20 dB. Hence the
Fig. 1. (Color online) (a) Schematic structure of metamaterials for generating OAM-carrying vector beams. (b) Geometric parameters: the radii are r i i 6.3 × 700 nm (i 0, 1) and the orientation angle is αφ lφ α0 (l 2, α0 0 as an example) with respect to the x axis. The rectangular aperture has a dimension of 600 × 140 nm. (c) Illustration of generating OAM-carrying vector beam (OAM charge number: 2, polarization order: 2).
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Fig. 2. (Color online) Spatial distributions of phase, power, and polarization of generated OAM-carrying vector beams [σ 1, left circularly polarized input beam; α0 0, along the direction of e1 φ].
electric field component E2 along the direction of e2 φ can be ignored. The fourth and fifth rows of Fig. 2 show spatial power distributions P x ; P y along the x and y axes, respectively. The sixth row of Fig. 2 shows calculated spatial polarization distribution, implying the generation of vector beams with a polarization order of l from 3 to −3. The obtained results shown in Fig. 2 confirm the successful generation of OAM-carrying vector beams using metamaterials. We then study the operation bandwidth. Metamaterials with orientation angle of αφ lφ α0 (l 1, 2, 3) are considered. Left circularly polarized light is adopted as the input excitation source. We use ER and purity to characterize the quality of the generated OAM-carrying vector beams. From the wavelength-dependent ER as shown in Fig. 3(a), one can clearly see the high-quality broadband generation of OAM-carrying vector beams ranging from 1000 to 2500 nm, i.e., from near-infrared to mid-infrared. For l 1 and 2, the ER is kept above 20 dB over a 1500 nm bandwidth (1000–2500 nm). For l 3, ER > 16 dB over a bandwidth of 1500 nm (1000–2500 nm) and ER > 20 dB over a bandwidth of 800 nm (1000–1800 nm) are achieved. Figure 3(b) shows
Fig. 3. (Color online) Wavelength-dependent (a) ER and (b) purity for generation of OAM-carrying vector beams. Insets in (b) show weight as functions of OAM charge number (left) and polarization order number (right) at 1550 nm.
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Fig. 5. (Color online) Dependence of ER on (a) initial orientation angle, (b) length, and (c) width of rectangular apertures. Fig. 4. (Color online) Spatial distributions of phase and polarization of generated OAM-carrying vector beams (σ −1, right circularly polarized input beam).
the wavelength-dependent purity for the OAM-carrying vector beam (l 3), which is larger than 0.85 over a bandwidth of 1500 nm (1000–2500 nm). The insets depict weight spectra as functions of OAM charge number and polarization order number at 1550 nm. High values of purity are achieved. We also investigate the dependence of the generated OAM-carrying vector beams on the polarization of input beams Ein 1 iσ T ; σ −1; 1 and the initial orientation angle α0 0; π∕2. We take metamaterials with orientation angle of αφ lφ α0 (l 1, 2, 3) as examples. Figure 4 shows spatial phase and polarization distributions of output beams when σ −1, α0 0 and σ −1, α0 π∕2. Compared with the spatial phase distributions shown in the first row of Fig. 2 σ 1; α0 0, it is found that the charge sign of OAM carried by the output beam is reversed as the polarization of the input beam changes. Such phenomenon can be easily explained by Eq. (2a), where the output beam contains a phase item of explσφ. In addition, by changing α0 from 0 [Eq. (2a): E1 along the direction of e1 φ] to π∕2 [Eq. (2b): E2 along the direction of e2 φ], we can get another group of OAM-carrying vector beams with a polarization rotation of π∕2. The middle two columns of Fig. 4 show the generation of OAM-carrying radially l 1; α0 0 and azimuthally l 1; α0 π∕2 polarized vector beams. The obtained results shown in Figs. 2 and 4 indicate that the proposed approach provides three degrees of freedom (l, σ, and α0 ) to flexibly generate different OAM-carrying vector beams. Beyond lower-order OAM-carrying radially and azimuthally polarized vector beams (l 1), it is also possible to generate higher-order OAM-carrying vector beams (l > 1) along the directions of e1 φ and e2 φ. We further consider the fabrication tolerance by evaluating the performance dependence of metamaterials on the offset of the initial orientation angle, length, and width of rectangular apertures at a wavelength of 1550 nm. Figure 5(a) shows the ER as a function of the initial orientation angle α0 of rectangular apertures. In the case of expected α0 0, when the value of α0 is out of the range of (−4, 6) deg, the ER tends to be deteriorated. Such phenomenon can be explained as follows: when α0 is offset, the polarization of the output beam is not perfectly along the direction of e1 φ. Hence, the electric field component E2 increases while E1 decreases, resulting in degradation of ER. Figures 5(b) and 5(c) show the ER as functions of the length L and
width W of rectangular apertures. It can be clearly seen that the ER (>25 dB) changes slightly (fluctuation:
